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\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  0.0 & +  6.00 x_{1} & +  5.00 x_{2} & -4.00 x_{3} & +  2.00 x_{4} & -9.00 x_{5}\\
 x_{7}   &  1.0  &   & +  7.00 x_{2} & +  6.00 x_{3} & +  5.00 x_{4} & + 10.00 x_{5}\\
 x_{8}   &  32.0 & -5.00 x_{1} & +  6.00 x_{2} & +  5.00 x_{3} & -5.00 x_{4} & -10.00 x_{5}\\
 x_{9}   &  0.0 & -7.00 x_{1} & -9.00 x_{2} & +  8.00 x_{3} &   & +  7.00 x_{5}\\
 x_{10}   &  -9.0 & -5.00 x_{1} & -9.00 x_{2} & +  7.00 x_{3} & +  8.00 x_{4} &   \\
\hline
z    &  0.0 & -3.00 x_{1} &   & +  4.00 x_{3} & -4.00 x_{4} & +  1.00 x_{5}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{5}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,10$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & -6.00 y_{6} &   & +  5.00 y_{8} & +  7.00 y_{9} & +  5.00 y_{10}\\
 y_{2}   &  1.0 & -5.00 y_{6} & -7.00 y_{7} & -6.00 y_{8} & +  9.00 y_{9} & +  9.00 y_{10}\\
 y_{3}   &  1.0 & +  4.00 y_{6} & -6.00 y_{7} & -5.00 y_{8} & -8.00 y_{9} & -7.00 y_{10}\\
 y_{4}   &  1.0 & -2.00 y_{6} & -5.00 y_{7} & +  5.00 y_{8} &   & -8.00 y_{10}\\
 y_{5}   &  1.0 & +  9.00 y_{6} & -10.00 y_{7} & + 10.00 y_{8} & -7.00 y_{9} &   \\
\hline
z    &  -0  &   & -1.00 y_{7} & -32.00 y_{8} &   & +  9.00 y_{10}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 2 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.625 & -7.25 y_{6} & -3.12 y_{7} & +  8.12 y_{8} & +  7.00 y_{9} & -0.62 y_{4}\\
 y_{2}   &  2.125 & -7.25 y_{6} & -12.62 y_{7} & -0.38 y_{8} & +  9.00 y_{9} & -1.12 y_{4}\\
 y_{3}   &  0.125 & +  5.75 y_{6} & -1.62 y_{7} & -9.38 y_{8} & -8.00 y_{9} & +  0.88 y_{4}\\
 y_{10}   &  0.125 & -0.25 y_{6} & -0.62 y_{7} & +  0.62 y_{8} &   & -0.12 y_{4}\\
 y_{5}   &  1.0 & +  9.00 y_{6} & -10.00 y_{7} & + 10.00 y_{8} & -7.00 y_{9} &   \\
\hline
z    &  1.125 & -2.25 y_{6} & -6.62 y_{7} & -26.38 y_{8} &   & -1.12 y_{4}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  2.25 & +  7.25 x_{1} & +  7.25 x_{2} & -5.75 x_{3} & +  0.25 x_{10} & -9.00 x_{5}\\
 x_{7}   &  6.625 & +  3.12 x_{1} & + 12.62 x_{2} & +  1.62 x_{3} & +  0.62 x_{10} & + 10.00 x_{5}\\
 x_{8}   &  26.375 & -8.12 x_{1} & +  0.38 x_{2} & +  9.38 x_{3} & -0.62 x_{10} & -10.00 x_{5}\\
 x_{9}   &  0.0 & -7.00 x_{1} & -9.00 x_{2} & +  8.00 x_{3} &   & +  7.00 x_{5}\\
 x_{4}   &  1.125 & +  0.62 x_{1} & +  1.12 x_{2} & -0.88 x_{3} & +  0.12 x_{10} &   \\
\hline
z    &  -1.125 & -1.62 x_{1} & -2.12 x_{2} & -0.12 x_{3} & -0.12 x_{10} & -1.00 x_{5}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  2.25 & +  7.25 x_{1} & +  7.25 x_{2} & -5.75 x_{3} & +  0.25 x_{10} & -9.00 x_{5}\\
 x_{7}   &  6.625 & +  3.12 x_{1} & + 12.62 x_{2} & +  1.62 x_{3} & +  0.62 x_{10} & + 10.00 x_{5}\\
 x_{8}   &  26.375 & -8.12 x_{1} & +  0.38 x_{2} & +  9.38 x_{3} & -0.62 x_{10} & -10.00 x_{5}\\
 x_{9}   &  0.0 & -7.00 x_{1} & -9.00 x_{2} & +  8.00 x_{3} &   & +  7.00 x_{5}\\
 x_{4}   &  1.125 & +  0.62 x_{1} & +  1.12 x_{2} & -0.88 x_{3} & +  0.12 x_{10} &   \\
\hline
z    &  -4.5 & -5.50 x_{1} & -4.50 x_{2} & +  7.50 x_{3} & -0.50 x_{10} & +  1.00 x_{5}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  2.25 & +  7.25 x_{1} & +  7.25 x_{2} & -5.75 x_{3} & +  0.25 x_{10} & -9.00 x_{5}\\
 x_{7}   &  6.625 & +  3.12 x_{1} & + 12.62 x_{2} & +  1.62 x_{3} & +  0.62 x_{10} & + 10.00 x_{5}\\
 x_{8}   &  26.375 & -8.12 x_{1} & +  0.38 x_{2} & +  9.38 x_{3} & -0.62 x_{10} & -10.00 x_{5}\\
 x_{9}   &  0.0 & -7.00 x_{1} & -9.00 x_{2} & +  8.00 x_{3} &   & +  7.00 x_{5}\\
 x_{4}   &  1.125 & +  0.62 x_{1} & +  1.12 x_{2} & -0.88 x_{3} & +  0.12 x_{10} &   \\
\hline
z    &  -4.5 & -5.50 x_{1} & -4.50 x_{2} & +  7.50 x_{3} & -0.50 x_{10} & +  1.00 x_{5}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{6} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  0.391304347826 & +  1.26 x_{1} & +  1.26 x_{2} & -0.17 x_{6} & +  0.04 x_{10} & -1.57 x_{5}\\
 x_{7}   &  7.26086956522 & +  5.17 x_{1} & + 14.67 x_{2} & -0.28 x_{6} & +  0.70 x_{10} & +  7.46 x_{5}\\
 x_{8}   &  30.0434782609 & +  3.70 x_{1} & + 12.20 x_{2} & -1.63 x_{6} & -0.22 x_{10} & -24.67 x_{5}\\
 x_{9}   &  3.13043478261 & +  3.09 x_{1} & +  1.09 x_{2} & -1.39 x_{6} & +  0.35 x_{10} & -5.52 x_{5}\\
 x_{4}   &  0.782608695652 & -0.48 x_{1} & +  0.02 x_{2} & +  0.15 x_{6} & +  0.09 x_{10} & +  1.37 x_{5}\\
\hline
z    &  -1.5652173913 & +  3.96 x_{1} & +  4.96 x_{2} & -1.30 x_{6} & -0.17 x_{10} & -10.74 x_{5}\\
\end{array}\]


 $ x_{1} $ enters and $ x_{4} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  2.45454545455 & -2.64 x_{4} & +  1.32 x_{2} & +  0.23 x_{6} & +  0.27 x_{10} & +  2.05 x_{5}\\
 x_{7}   &  15.7272727273 & -10.82 x_{4} & + 14.91 x_{2} & +  1.36 x_{6} & +  1.64 x_{10} & + 22.27 x_{5}\\
 x_{8}   &  36.0909090909 & -7.73 x_{4} & + 12.36 x_{2} & -0.45 x_{6} & +  0.45 x_{10} & -14.09 x_{5}\\
 x_{9}   &  8.18181818182 & -6.45 x_{4} & +  1.23 x_{2} & -0.41 x_{6} & +  0.91 x_{10} & +  3.32 x_{5}\\
 x_{1}   &  1.63636363636 & -2.09 x_{4} & +  0.05 x_{2} & +  0.32 x_{6} & +  0.18 x_{10} & +  2.86 x_{5}\\
\hline
z    &  4.90909090909 & -8.27 x_{4} & +  5.14 x_{2} & -0.05 x_{6} & +  0.55 x_{10} & +  0.59 x_{5}\\
\end{array}\]


 $ x_{2} $ enters and Unbounded Dictionary!


 $ x_{2} $ enters and Unbounded Dictionary!
\end{document}
